We know that two quantum states, can not get entangled via local classical interactions/communication (LOCC). However, do two quantum states locally interacting via quantum interactions always get entangled?
Suppose, two Quantum systems, initially in the unentangled state, interact locally with a mediator, and no entanglement is generated between the two because of interaction, can we unambiguously determine that the mediator was classical in nature?
However, do two quantum states locally interacting via quantum interactions always get entangled?
No. I can give you an example using the $CNOT$ gate for qubits which is defined by the following relation:
$$ CNOT |0\rangle |0\rangle = |0 \rangle |0\rangle $$ $$ CNOT |0\rangle |1\rangle = |0 \rangle |1\rangle $$ $$ CNOT |1\rangle |0\rangle = |1 \rangle |1\rangle $$ $$ CNOT |1\rangle |1\rangle = |1 \rangle |0\rangle $$
i.e. the second qubit is flipped if the first one is in state $|1\rangle$. Clearly none of the above states is entangled after operating with CNOT.This is because the state can be written in the form $|\cdot\rangle_{q_1}|\cdot\rangle_{q_2}$ where $q_1$ and $q_2$ are qubits 1 and 2.
Now, look at the not-entangled state $$\frac{1}{\sqrt 2}(|0\rangle + |1\rangle)\otimes |1\rangle = \frac{1}{\sqrt 2} (|0\rangle|1\rangle + |1\rangle|1\rangle)$$
after acting with $CNOT$ we obtain: $$CNOT \frac{1}{\sqrt 2} (|0\rangle|1\rangle + |1\rangle|1\rangle) = \frac{1}{\sqrt 2} (|0\rangle|1\rangle + |1\rangle |0\rangle)$$
using the defining relations of the CNOT gate. The state $\frac{1}{\sqrt 2} (|0\rangle|1\rangle + |1\rangle |0\rangle)$ is called Bell state and it is a maximally entangled state. Hence, you can see that the final level of entanglement also depends on the initial state that experiences the interaction.
After doing some more reading, I came across the concept of entangling power. For a gate $U$, the entangling power $K$ is defined as [1]:
$$K_{E}(U)=\max _{|\phi\rangle,|\psi\rangle} E(U(|\phi\rangle|\psi\rangle))$$
This quantity is the maximum attainable entanglement $E$ maximised over all states $|\phi\rangle|\psi\rangle$.
[1] https://iopscience.iop.org/article/10.1088/1751-8121/aad7cb/pdf
